Effects of Torsion on Electromagnetic Fields

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636 Brazilian Journal of Physics,vol.35,no.3A,September,2005
Effects of Torsion on Electromagnetic Fields
La
´
ercio Dias
Laborat
´
orio de F
´
sica Te
´
orica e Computacional,Departamento de F
´
sica,
Universidade Federal de Pernambuco,50670-901,Recife,PE,Brazil
and Fernando Moraes
Laborat
´
orio de F
´
sica Te
´
orica e Computacional,Departamento de F
´
sica,
Universidade Federal de Pernambuco,50670-901,Recife,PE,Brazil and
Departamento de F
´
sica,CCEN,Universidade Federal da Para
´
ba,
Cidade Universit
´
aria,58051-970 Jo

ao Pessoa,PB,Brazil
Received on 14 February,2005
In this work,we investigate the effects of torsion on electromagnetic elds.As a model spacetime,en-
dowed with both curvature and torsion,we choose a generalization of the cosmic string,the cosmic dislocation.
Maxwell's equations in the spacetime of a cosmic dislocation are then solved,considering both the case of a
static,uniform,charge distribution along the string,and the case of a constant current owing through the string.
We nd that the torsion associated to the defect affects only the magnetic eld whereas curvature affects both
electric and magnetic elds.Moreover,the magnetic eld is found to spiral up around the defect axis.
I.INTRODUCTION
The study of electromagnetism in a curved background
has very important astrophysical implications as for exam-
ple helping the understanding of the signals received from
neutron stars and maybe also from black holes.Electromag-
netic processes near such objects certainly will have general-
relativistic effects.After the generalization of Einstein's grav-
itational theory to include torsion,done by Hehl and cowork-
ers [1],one might ask what are the effects of torsion on elec-
tromagnetic elds.In this work we study a simple but illus-
trative case:the electromagnetic eld produced by a cyllindri-
cally symmetric source coincident with a topological line de-
fect that carries both curvature and torsion.
Topological structures like domain walls,strings and
monopoles may have been produced by phase transitions in-
volving spontaneous symmetry breaking in the early uni-
verse [2].Such defects are associated to curvature singu-
larities [3] and are solutions to Einstein's eld equations.
Although astronomical observations keep indicating that the
macroscopic geometry of the universe is Riemannian it is
possible that torsion may appear near curvature singularities
[4].Line defects containing torsion,like dislocations,appear
within Einstein-Cartan-Sciama-Kibble gravitation theory [1]
in Riemann-Cartan spacetime U
4
.
We consider the cosmic dislocation [5] spacetime whose
metric is given by
ds
2
=¡dt
2
+dr
2
+
2
r
2
d
2
+(dz + d )
2
;(1)
in cylindrical coordinates.The parameter  is associated with
the angular decit of a cosmic string spacetime.The values
of  are restricted to the interval 0 < <1,since the linear
density of mass of a cosmic string,given by µ =(1¡ )=4G,
must be positive.The parameter  is related to the torsion
associated to the defect.For dislocations in solid state physics
 is related to the Burgers vector
~
b by  =
b
2
.
This topological defect carries both torsion and curvature,
both appearing as conical singularities on the z-axis.The only
nonzero component of the torsion tensor in this case is given
by the two-form[6]
T
z
=2
2
(r)dr ^d;(2)
where 
2
(r) is the two-dimensional delta function.Analo-
gously,the nonvanishing components of the curvature two-
formare [6]
R
r

=¡R

r
=2
(1¡ )


2
(r)dr ^d:(3)
This study intends to show how the changes introduced in
the geometrical structure of spacetime by a topological line
defect affect the solutions of Maxwell's equations.We are
interested in the following two cases involving a cosmic dis-
location:(i) the defect carries a density of charge and (ii) it
carries a current.Asimilar problemwas handled by M.F.A.da
Silva et al.[7,8],who calculated the magnetostatic eld due
to an electric current placed in the gravitational background
of a rotating cosmic string.In their work,torsion comes from
rotation,thus coupling time to the angular coordinate .Here,
torsion comes fromthe coupling between  and z,as it is clear
from Eq.(1).Other cases,with spherical symmetry,have
appeared in the literature.For example,the electrostatic eld
and the potential of a point charge in the Schwartzschild met-
ric obtained by Linet [9] and the magnetostatic eld of a loop
current around a black hole obtained by Petterson [10].Re-
lated to these problems is the question of the self-force on
electric and magnetic sources in the presence of topological
defects,the object of much attention in recent years [11]-[19].
We restrict ourselves to the study of the generation of elec-
tric and magnetic elds by static sources.To facilitate the
calculations,we consider the approximation [10] where the
electromagnetic eld is taken as a weak perturbation on the
spacetime metric.Thus the inuence of the metric on the
electromagnetic eld is much stronger than the inuence of
the electromagnetic eld on the metric.In this approxima-
tion,the task of solving Einstein-Maxwell equations reduces
to solving Maxwell equations in covariant form.
La
´
ercio Dias and Fernando Moraes 637
This work is organized as follows.In Section II,we derive
Maxwell equations in the cosmic dislocation spacetime.In
Section III we solve them for the electrostatic eld generated
by a line of charge.We nd that there is no effect of torsion
on the electric eld.On the other hand curvature amplies
it.In Section IV we calculate the magnetostatic eld of a line
current in the spacetime of the cosmic dislocation,nding an
interesting effect:the peculiarities of the metric give rise to a
z-component of the eld.Finally,in Section V we we present
our concluding remarks.We observe that in this paper we use
geometrical units.
II.MAXWELL EQUATIONS IN THE COSMIC
DISLOCATION SPACETIME
We start by writting Maxwell equations using differential
forms:
dF =0 (4)
and
?d?F =J;(5)
where
F =
1
2
F
µ
dx
µ
^dx

=B+E^dt (6)
is the Faraday two-formand J is the current density one-form
given by
J =¡ dt +J
r
dr +J

d +J
z
dz:(7)
In Eq.(6) the magnetic eld is represented by the two-form B
and the electric eld by the one-form E.
The Faraday two-form in terms of its components is there-
fore
F =F
 z
d ^dz +F
zr
dz ^dr +F
r
dr ^d
+F
rt
dr ^dr +F
 t
d ^dt +F
zt
dz ^dt:(8)
Eqs.(6) and (8) imply that
B =F
 z
d ^dz +F
zr
dz ^dr +F
r
dr ^d (9)
and
E^dt =F
rt
dr ^dt +F
 t
d ^dt +F
zt
dz ^dt;(10)
which leads to
E
r
=F
rt
E

=F
 t
E
z
=F
zt
;(11)
where
E ´E
r
dr +E

d +E
z
dz:(12)
We observe that the electric eld vector components
(E
r
;E

;E
z
) are related to the one-form E components
(E
r
;E

;E
z
) by the metric in the usual way contravariant and
covariant vector components are related.In the same way,the
magnetic eld vector components (B
r
;B

;B
z
) are related to a
magnetic eld one-form B
1
components (B
r
;B

;B
z
).Never-
theless,the two-form B is related to B
1
by the Hodge?oper-
ation:
?B =B
1
^dt:(13)
Applying the Hodge?operator on Eq.(9) we obtain
?B =
µ

2
r
2
+
2
 r
F
zr
+

 r
F
r

d ^dt
+
µ
1
 r
F
r
+

 r
F
zr

dz ^dt +
F
 z
 r
dr ^dt:(14)
Therefore,we identify the components of the magnetic eld
one-formB
1
B
r
=
1
 r
F
 z
B

=

2
r
2
+
2
 r
F
zr
+

 r
F
r
(15)
B
z
=
1
 r
F
r
+

 r
F
zr
:
Now,applying the Hodge?operator on Eq.(10) we obtain
?(E^dt) = ¡  rF
rt
d ^dz
¡
µ
1
 r
F
 t
¡

 r
F
zt

dz ^dr (16)
¡
µ

2
r
2
+
2
 r
F
zt
¡

 r
F
 t

dr ^d:
Finally,using Eqs.(6),(11),(14),(15) and (16),we obtain
?d?F =f
1
 r
 B
z

¡
1
 r
 B

 z
¡
 E
r
 t
gdr
+ f

 r
 B

 r
¡

 r
 B
r

+

2
r
2
+
2
 r
 B
r
 z
¡

2
r
2
+
2
 r
 B
z
 r
¡
 E

 t
gd
+ f
1
 r
 B

 r
¡
1
 r
 B
r

+

 r
 B
r
 z
¡

 r
 B
z
 r
¡
 E
z
 t
gdz
¡ f
1
r

 r
(rE
r
) +


(
1

2
r
2
E

¡


2
r
2
E
z
)
+
1

2
r
2

 z
[(
2
r
2
+
2
)E
z
¡ E

]gdt:(17)
Care should be taken in interpreting the contravariant com-
ponents of the elds since the metric (1) is associated to a non-
orthonormal basis (~e
t
;~e
r
;~e

;~e
z
),where g
µ
=~e
µ
¢~e

.There-
fore we need to relate the components of the electric and mag-
netic eld one-forms to the respective vectors in a normalized
basis,such that in the no defect limit we recover the elds
generated by a line source in at spacetime.The new basis
(~e

t
;~e
r
;~e


;~e
z
) is simply obtained by
~e
µ
=
~e
µ
p
g
µµ
:(18)
638 Brazilian Journal of Physics,vol.35,no.3A,September,2005
The components of a generic 1-form A ´ A
r
dr +A

d +
A
z
dz are related to the components of the equivalent vector
~
A=A
r
~e
r
+A


~e


+A
z
~e
z
,expressed in the normalized (but non-
orthogornal) basis,by:
A
r
=A
r
(19)
A

=
q

2
r
2
+
2
A


+ A
z
(20)
A
z
=

p

2
r
2
+
2
A


+A
z
:(21)
After some algebraic manipulations we nally obtain Eq.
(5) in terms of the components of the electric,magnetic and
current density vectors
1
r

 r
(rE
r
) +
1
p

2
r
2
+
2
 E



+
 E
z
 z
=;(22)
1
 r
Ã

p

2
r
2
+
2


¡
q

2
r
2
+
2

 z
!
B


+
1
 r
µ


¡

 z

B
z
=J
r
+
 E
r
 t
;(23)
p
(
2
r
2
+
2
)
 r
"
 B
r
 z
¡

 r
Ã
B
z
+

p

2
r
2
+
2
B


!#
=J


+
 E


 t
;(24)
1
 r
·

 r
µ
q

2
r
2
+
2
B


+ B
z

¡
 B
r

¸
=J
z
+
 E
z
 t
:(25)
Notice that Eq.(22) corresponds to Gauss lawand that Eqs.
(23 - 25) correspond to Amp

ere-Maxwell law.
In a similar way,Eq.(4) leads to
1
r

 r
(rB
r
) +
1
p

2
r
2
+
2
 B



+
 B
z
 z
=0;(26)
1
 r
Ã

p

2
r
2
+
2


¡
q

2
r
2
+
2

 z
!
E


+
1
 r
µ


¡

 z

E
z
+
 B
r
 t
=0;(27)
p

2
r
2
+
2
 r
"
 E
r
 z
¡

 r
Ã
E
z
+

p

2
r
2
+
2
E


!#
+
 B


 t
=0;(28)
1
 r
·

 r
µ
q

2
r
2
+
2
E


+ E
z

¡
 E
r

¸
+
 B
z
 t
=0:(29)
Now,Eq.(26) describes the absence of magnetic monopoles
and Eqs.(27 - 29) correspond to Faraday law.
III.ELECTRIC FIELD OF THE LINE CHARGE
In this section we briey discuss the case of a uniform line
of charge coincident with the cosmic dislocation.In this case,
the charge density is described by
 (r) =

2
 (r)
r
;(30)
where  is the linear charge density.The presence of  in this
expression is due to the change in the volume element caused
by the string metric.
The symmetries of the problem suggest that E
r
=
E
r
(r);E


=E


(r) and E
z
=E
z
(r).Eqs.(27-29) imply readily
that
E


(r) =E
z
(r) =0 (31)
and Eq.(22) gives
E
r
(r) =

2
1
r
:(32)
This result might be explained by a simple argument based
on the electric eld lines,as follows.The process of creat-
ing the defect involves cutting out a wedge of space,which
leaves less volume for the eld lines to spread through.This
increases the density of eld lines therefore corresponding to
an amplication of the electric eld amplitude.This should
be compared to the amplication found in the magnetostatic
eld of a current-carrying cosmic string [7]).
IV.MAGNETIC FIELD OF THE LINE CURRENT
Now we treat the case where a current ows along the de-
fect.The important equations now are (23 - 25).Here the
symmetry suggests that the nonvanishing components of the
magnetic eld are B


= B


(r) and B
z
= B
z
(r).With this,in
the region r >0,Eqs.(24) and (25) turn into
d
dr
Ã
B
z
+

p

2
r
2
+
2
B


!
=0 (33)
d
dr
µ
q

2
r
2
+
2
B


+ B
z

=0:(34)
We have thus a coupled set of equations of very simple so-
lution:
B


(r) =k
1
p

2
r
2
+
2

2
r
2
(35)
and
B
z
(r) =k
2
¡

2
2
r
2
;(36)
where k
1
and k
2
are integration constants.In order to deter-
mine these constants we withdraw the defect by setting  =1
La
´
ercio Dias and Fernando Moraes 639
FIG.1: -component of the magnetic eld:¤in at spacetime (  =
1 and  =0),± in the cosmic string spacetime ( =0:5 and  =0),
¦ in the cosmic dislocation spacetime ( =0:5 and  =1)
and  =0.Thus,we recover the magnetic eld of a line cur-
rent in at spacetime:
B


 =1; =0
(r) =
I
2 r
;(37)
B
z
 =1; =0
(r) =0;(38)
where I is the electric current.Hence,we have k
1
=
I
2
and
k
2
= 0.Substituting this into Eqs.(35) and (36),we nally
get
B


(r) =
I
2
p

2
r
2
+
2

2
r
2
;(39)
B
z
(r) =¡
I
2


2
r
2
:(40)
The coupling between the angular and the z coordinates
brings about an unexpected component of the magnetic eld,
which vanishes properly in the no torsion limit !0.In Fig.
1 it is shown the  -component of the magnetic eld in a few
illustrative cases.In what follows we take a closer look at the
magnetic eld lines in this torsioned space.
In order to nd the integral curves (magnetic eld lines) of
our vector (magnetic) eld we need to solve the parametric
systembelow:
r(t) = B
r
(r;;z)

 (t) = B

(r;;z) (41)
z(t) = B
z
(r;;z);
where t is a parameter.Since B
r
=0,B

=B


p
g

and B
z
=
B
z
(see Eq.(18)) and with Eqs.(39) and (40) we have
r(t) = 0

 (t) =
I
2
1

2
r
2
(42)
z(t) = ¡
I
2


2
r
2
;
whose solution is
r(t) = r
0
 (t) =
I
2
2
r
2
0
t +
0
(43)
z(t) = ¡
I
2
2
r
2
0
t +z
0
;
where r
0
;
0
;z
0
are integration constants.
It is clear that the above set of equations describes a helix.
V.CONCLUDINGREMARKS
In this work we investigated the inuence of the torsion
and curvature of a topological defect on electromagnetic elds
generated by a line source coinciding with the defect.Torsion
affects the magnetic eld whereas curvature affects both elec-
tric and magnetic elds,but in different ways.Also,torsion
forces the magnetic eld lines to spiral up along the defect
axis.
Acknowledgments
We are grateful to CNPq,FINEP(PRONEX) and CAPES
(PROCAD) for partial support of this work.
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